Finite Group Databasebeta

← #5927

Subgroups — #5927

Order 192.

140 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
12−1+4⋊C3C211yesmaximal, characteristic, complemented
2#985C211yesmaximal, characteristic, complemented
3#984C211yesmaximal, characteristic, complemented
4#983C212yesmaximal, complemented
5C2×#226C212yesmaximal, complemented
6#578—31nomaximal, Sylow
7#237—41nomaximal
8#41⋊C3C2211yescharacteristic
9C2×SL(2,𝔽3)C2212yes—
10GL(2,𝔽3)C2211yescharacteristic
11#226C2211yescharacteristic
12#226C2212yes—
132−1+4—31no—
142−1+4D311yescharacteristic, complemented
15#137—31no—
16#137—32no—
17#137—31no—
18#137—32no—
19#135—32no—
20#135—31no—
21C2×Q16—31no—
22C2×Q16—32no—
23#131—31no—
24C3×D4—41no—
25#67—42no—
26C2×Dic3—42no—
27C4×D3—41no—
28Dic6—41no—
29SL(2,𝔽3)C2311yescharacteristic
30#41—31no—
31#41—32no—
32#41—32no—
33#41—31no—
34#41—62no—
35#41—61no—
36#41C2×D311yescharacteristic
37C2×Q8—31no—
38C2×Q8—32no—
39C2×Q8—31no—
40C2×Q8—61no—
41C2×Q8C2×D312yes—
42Q16—32no—
43Q16—32no—
44Q16—32no—
45Q16—31no—
46Q16—32no—
47SD16—32no—
48SD16—31no—
49SD16—32no—
50SD16—31no—
51D8—31no—
52M16—32no—
53M16—31no—
54C2×C8—32no—
55C2×C8—31no—
56C22×C3—42no—
57C2×D3—41no—
58C3×C4—41no—
59Dic3—42no—
60Dic3—41no—
61Q8—32no—
62Q8—31no—
63Q8—32no—
64Q8—31no—
65Q8—62no—
66Q8—61no—
67Q8C22×D311yescharacteristic
68D4—31no—
69D4—32no—
70D4—31no—
71D4—61no—
72D4—62no—
73D4S411yescharacteristic
74C2×C4—32no—
75C2×C4—31no—
76C2×C4—61no—
77C2×C4—62no—
78C2×C4—61no—
79C2×C4—62no—
80C2×C4—61no—
81C8—31no—
82C8—32no—
83C8—31no—
84C2×C3—41no—
85C2×C3—82no—
86D3—81no—
87C22—31no—
88C22—61no—
89C22C2×S412yes—
90C4—31no—
91C4—32no—
92C4—62no—
93C4—61no—
94C4C2×S411yescharacteristic
95C3—41noSylow
96C2—22no—
97C2—61no—
98C2—121no—
99C2C22×S411yescharacteristic
subgroupG/H
center, Frattini, socleC2C22×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23
Fitting2−1+4D3
hypercenterD4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. 2−1+4⋊C3C2semidirect
2. #985C2semidirect
3. #984C2semidirect
4. #983C2semidirect
5. C2×#226C2semidirect
14. 2−1+4D3semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
2−1+4⋊C32
#9852
#9842
#9832
C2×#2262
#5783
#2374